To solve the problem of determining the height of the pole given the distance from the pole and the angle of elevation, we can follow these steps:
1. Understand the relationship between the components of the right triangle:
- The distance between the person and the telephone pole forms the adjacent side of a right triangle.
- The height of the pole forms the opposite side of the triangle.
- The angle of elevation from the ground to the top of the pole is given as [tex]\( 30^\circ \)[/tex].
2. Use the tangent function:
The tangent of an angle in a right triangle is equal to the ratio of the length of the opposite side to the length of the adjacent side. Mathematically, this can be expressed as:
[tex]\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\][/tex]
3. Substitute the given values:
Here, [tex]\(\theta = 30^\circ\)[/tex] and the adjacent side is 36 feet. The relationship using the tangent function becomes:
[tex]\[
\tan(30^\circ) = \frac{\text{height of the pole}}{36 \text{ ft}}
\][/tex]
4. Solve for the height of the pole:
We know that [tex]\(\tan(30^\circ) = \frac{1}{\sqrt{3}}\)[/tex]. Thus, our equation can be rewritten as:
[tex]\[
\frac{1}{\sqrt{3}} = \frac{\text{height of the pole}}{36}
\][/tex]
Multiplying both sides of the equation by 36 to isolate the height, we get:
[tex]\[
\text{height of the pole} = 36 \times \frac{1}{\sqrt{3}}
\][/tex]
To simplify [tex]\( 36 \times \frac{1}{\sqrt{3}} \)[/tex], we can rationalize it. Multiplying the numerator and the denominator by [tex]\(\sqrt{3}\)[/tex], we get:
[tex]\[
\text{height of the pole} = 36 \times \frac{\sqrt{3}}{3}
\][/tex]
Simplifying this, we find:
[tex]\[
\text{height of the pole} = 12 \sqrt{3}
\][/tex]
Therefore, the height of the pole is [tex]\( 12 \sqrt{3} \)[/tex] feet.
